A Review of Certain Differential Equations research that is Relevant to Differential Difference Equations: Exploring the Advancements in Differential Equations for Differential Difference Equations Study

Abstract

Long before computers, differential equations were used to model and analyze a wide range ofphysical, biological, and engineering systems. However, with the development of computing technologiesand the need to take discrete events into account, researchers' focus has shifted to a subset ofequations known as differential difference equations (DDEs). The purpose of this work is to survey thestate of knowledge on differential equations and discuss how these advancements may be used to thestudy of differential difference equations. The study starts out by providing a primer on ODEs and PDEs,the two types of differential equations most often used to represent continuous systems, and theirfundamental ideas. It then goes on to explain how DDEs came to be as an extension of ODEs, whichallowed for delayed effects and discrete time steps. The many uses of dynamical systems equations(DDEs) are discussed. These use cases range from population dynamics to physiological systems tocontrol theory.